ACTA AERONAUTICAET ASTRONAUTICA SINICA >
Unstructured deforming mesh and discrete geometric conservation law
Received date: 2016-01-16
Revised date: 2016-05-03
Online published: 2016-05-10
Supported by
National Natural Science Foundation of China (91541117)
A popular method for simulating unsteady flow of fluid-structure interaction or multi-body separation is finite volume method based on arbitrary Lagrangian-Eulerian (ALE) equations, which involves mesh deformation and discrete geometric conservation law. The fundamental principle, state of the art and boundaries of validity of mesh deformation are reviewed following 5 categories of constructing ideas, e.g., physics analogy, ellipse smoothing, interpolation, moving submesh approach (MSA) and hybrid method. A hybrid method which combines the benefits of MSA and radial basis functions (RBFs) interpolation is proved to be robust and efficient via several numerical examples. After the concept of discrete geometric conservation law (DGCL) is introduced, the intrinsic mechanism of DGCL is analyzed through 2D model, which is the inequality between volume increment and the volume sweeping by the surfaces enclosing of mesh cell. The different implementations of DGCL which could be mainly categorized as area correction, area correction via assuming velocity of surface, velocity correction and volume correction are surveyed and their range of application and the existing problems are discussed. We found that the proposed volume correction method can satisfy the fluid-structure interface condition, and is also appropriate for multi-step temporal discrete schemes.
LIU Jun , LIU Yu , CHEN Zedong . Unstructured deforming mesh and discrete geometric conservation law[J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2016 , 37(8) : 2395 -2407 . DOI: 10.7527/S1000-6893.2016.0141
[1] 杨永健, 张鲁民. 应用重叠网格技术求解复杂组合体无粘流场[J]. 空气动力学报, 1991, 31(1):51-57. YANG Y J, ZHAMG L M. Numerical simulation of inviscid flow over a complicated body using an overlapping grid technique[J]. Acta Aerodynamica Sinica, 1991, 31(1):51-57(in Chinese).
[2] 范晶晶, 阎超, 张辉. 重叠网格洞面优化技术的改进与应用[J]. 航空学报, 2010, 31(6):1127-1133. FAN J J, YAN C, ZHANG H. Improvement of hole-surface optimization technique in overset grids and Its application[J]. Acta Aeronautica et Astronautica Sinica, 2010, 31(6):1127-1133(in Chinese).
[3] 张培红, 王明, 邓有奇, 等. 武器分离及舱门开启过程数值模拟研究[J]. 空气动力学学报, 2013, 31(3):277-281. ZHANG P H, WANG M, DENG Y Q, et al. Numerical simulation of store separation and door operation[J]. Acta Aerodynamica Sinica, 2013, 31(3):277-281(in Chinese).
[4] 肖中云, 江雄, 牟斌, 等. 并行环境下外挂物动态分离过程的数值模拟[J]. 航空学报, 2010, 31(8):1509-1516. XIAO Z Y, JIANG X, MOU B, et al. Numerical simulation of dynamic process of store separation in parallel environment[J]. Acta Aeronautica et Astronautica Sinica, 2010, 31(8):1509-1516(in Chinese).
[5] 刘学强, 伍贻兆, 夏健. 多重网格法在非结构网格中的应用[J]. 计算物理, 2002, 19(4):357-361. LIU X Q, WU Z Y, XIA J. Application of the multigrid method in unstructured meshes[J]. Chinese Journal of Computational Phyfics, 2002, 19(4):357-361(in Chinese).
[6] 李盾, 何跃龙, 纪楚群. 多体分离数值模拟研究与应用[C]//北京力学会第19届学术年会论文集. 北京:北京力学会, 2013:121-122. LI D, HE Y L, JI C Q. Numerical simulation research and application of multi-body separation[C]//The Beijing Society of Theoretical and Applied Mechanics 19th Annual Conference Proceedings. Beijing:The Beijing Society of Theoretical and Applied Mechanics, 2013:121-122(in Chinese).
[7] 何跃龙, 李盾, 刘晓文. 非结构直角网格动网格方法[C]//北京力学会第18届学术年会论文集. 北京:北京力学会, 2012:19-20. HE Y L, LI D, LIU X W. Dynamic unstructured cartesian grid method[C]//The Beijing Society of Theoretical and Applied Mechanics 18th Annual Conference Proceedings. Beijing:The Beijing Society of Theoretical and Applied Mechanics, 2012:19-20(in Chinese).
[8] 刘君, 白晓征, 郭正. 非结构动网格计算方法-及其在包含运动界面的流场模拟中的应用[M]. 长沙:国防科技大学出版社, 2009. LIU J, BAI X Z, GUO Z. Dynamic unstructured grid method and its application in simulation of unsteady flows involving moving boundaries[M]. Changsha:National University of Defense Technology Press, 2009(in Chinese).
[9] BATINA J T. Unsteady euler airfoil solutions using unstructured dynamic meshes[J]. AIAA Journal, 1990, 28(8):1381-1388.
[10] FARHAT C, DEGAND C, KOOBUS B, et al. Torsional springs for two-dimensional dynamic unstructured fluid meshes[J]. Computer Methods in Applied Mechanics and Engineering, 1998, 163(1):231-245.
[11] DEGAND C, FARHAT C. A three-dimensional torsional spring analogy method for unstructured dynamic meshes[J]. Computers & Structures, 2002, 80(3):305-316.
[12] BLOM F J. Considerations on the spring analogy[J]. International Journal for Numerical Methods in Fluids, 2000, 32(6):647-668.
[13] 郭正, 刘君, 瞿章华. 非结构动网格在三维可动边界问题中的应用[J]. 力学学报, 2003, 35(2):140-146. GUO Z, LIU J, QU Z H. Dynamic unstructured grid method with applications to 3D unsteady flows involving moving boundaries[J]. Acta Mechanica Sinica, 2003, 35(2):140-146(in Chinese).
[14] BOTTASSO C L, DETOMI D, SERRA R. The ball-vertex method:A new simple spring analogy method for unstructured dynamic meshes[J]. Computer Methods in Applied Mechanics and Engineering, 2005, 194(39):4244-4264.
[15] ACIKGOZ N, BOTTASSO C L. A unified approach to the deformation of simplicial and non-simplicial meshes in two and three dimensions with guaranteed validity[J]. Computers & Structures, 2007, 85(11):944-954.
[16] TEZDIUAR T E, BEHR M, MITTAL S, et al. Computation of unsteady incompressible flows with the finite element methods:Space-time formulations, iterative strategies and massively parallel implementations[J]. Asme Pressure Vessels Piping Div Publ PVP, 1992, 246(1):7-24.
[17] JOHNSON A A, TEZDUYAR T. Mesh update strategies in parallel finite element computations of flow problems with moving boundaries and interfaces[J]. Computer Methods in Applied Mechanics and Engineering, 1994, 119(1-2):73-94.
[18] JOHNSON A A, TEZDUYAR T E. Advanced mesh generation and update methods for 3D flow simulations[J]. Computational Mechanics, 1999, 23(2):130-143.
[19] STEIN K, TEZDUYAR T, BENNEY R. Mesh moving techniques for fluid-structure interactions with large displacements[J]. Journal of Applied Mechanics, 2003, 70(1):58-63.
[20] STEIN K, TEZDUYAR T E, BENNEY R. Automatic mesh update with the solid-extension mesh moving technique[J]. Computer Methods in Applied Mechanics and Engineering, 2004, 193(21):2019-2032.
[21] CHIANDUSSI G, BUGEDA G, ONATE E. A simple method for automatic update of finite element meshes[J]. Computer Methods in Applied Mechanics and Engineering, 2000, 16(1):1-19.
[22] XU Z, ACCORSI M. Finite element mesh update methods for &uid-structure interaction simulations[J]. Finite Elements in Analysis and Design, 2004, 40(9):1259-1269.
[23] SMITH R W. A PDE-based mesh update method for moving and deforming high reynolds number meshes:AIAA-2011-0472[R]. Reston:AIAA, 2011.
[24] SMITH R W, WRIGHT J A. A Classical elasticity-based mesh update method for moving and deforming meshes:AIAA-2010-0164[R]. Reston:AIAA, 2010.
[25] YANG Z, MAVRIPLIS D J. Unstructured dynamic meshes with higher-order time integration schemes for the unsteady Navier-Stokes equations:AIAA-2005-1222[R]. Reston:AIAA, 2005.
[26] KARMAN S L, ANDERSON W K, SAHASRABUDHE M. Mesh generation using unstructured computational meshes and elliptic partial differential equation smoothing[J]. AIAA Journal, 2006, 44(6):1277-1286.
[27] KENNON S R, MEYERING J M, BERRY C W. Geometry based Delaunay tetrahedralization and mesh movement strategies for multi-body CFD:AIAA-1992-4575[R]. Reston:AIAA, 1992.
[28] 陈炎, 曹树良, 梁开洪, 等. 基于温度体模型的动网格生成方法及在流固耦合振动中的应用[J]. 振动与冲击, 2010, 29(4):1-5. CHEN Y, CAO S L, LIANG K H, et al. A new dynamic grids based on temperature analogy and its application in vibration engineering with fluid-solid interaction[J]. Journal of Vibration and Shock, 2010, 29(4):1-5(in Chinese).
[29] 陈炎, 曹树良, 梁开洪, 等. 温度体动网格模型中控制参数的研究[J]. 计算物理, 2010, 27(3):396-402. CHEN Y, CAO S L, LIANG K H, et al. Parameter control in temperature analogy method[J]. Chinese Journal of Computational Phyfics, 2010, 27(3):396-402(in Chinese).
[30] 陈炎, 张勤昭, 曹树良. 温度体动网格方法的旋转变形能力[J]. 排灌机械工程学报, 2012, 30(4):447-451. CHEN Y, ZHANG Q Z, CAO S L. Rotational deformability of temperature analogy method[J]. Journal of Drainage and Irrigation Machinery Engineering, 2012, 30(4):447-451(in Chinese).
[31] 陈炎, 张勤昭, 曹树良, 等. 基准温度分布动网格生成方法的研究及应用[J]. 北京理工大学学报, 2012, 32(9):900-904. CHEN Y, ZHANG Q Z, CAO S L, et al. A new method of dynamic grid generation based on reference temperature distribution[J]. Transactions of Beijing Institute of Technology, 2012, 32(9):900-904(in Chinese).
[32] THOMPSON J F, WARSI Z U A, MASTIN C W. Numerical grid generation:Foundations and applications[M]. Amsterdam:North-holland, 1985.
[33] LÖHNER R, YANG C. Improved ALE mesh velocities for moving bodies[J]. Communications in Numerical Methods in Engineering, 1996, 12(10):599-608.
[34] LOMTEV I, KIRBY R M, KARNIADAKIS G E. A discontinuous Galerkin ALE method for compressible viscous flows in moving domains[J]. Journal of Computational Physics, 1999, 155(1):128-159.
[35] CALDERER R, MASUD A. A multiscale stabilized ALE formulation for incompressible flows with moving boundaries[J]. Computational Mechanics, 2010, 46(1):185-197.
[36] HUSSAIN M, ABID M, AHMAD M, et al. A parallel implementation of ALE moving mesh technique for FSI problems using openMP[J]. International Journal of Parallel Programming, 2011, 39(6):717-745.
[37] MASUD A, BHANABHAGVANWALA M, KHURRAM R A. An adaptive mesh rezoning scheme for moving boundary flows and fluid-structure interaction[J]. Computers & Fluids, 2007, 36(1):77-91.
[38] MASUD A, HUGHESB T J R. A space-time Galerkin/least-squares finite element formulation of the Navier-Stokes equations for moving domain problems[J]. Computer Methods in Applied Mechanics and Engineering, 1997, 146(1):91-126.
[39] HELENBROOK B T. Mesh deformation using the biharmonic operator[J]. International Journal for Numerical Methods in Engineering, 2003, 56(7):1007-1021.
[40] WANG Q, HU R. Adjoint-based optimal variable stiffness mesh deformation strategy based on bi-elliptics equations[J]. International Journal for Numerical Methods in Engineering, 2012, 90(5):659-670.
[41] YIGIT S, FER M S, HECK M. Grid movement techniques and their influence on laminar fluid-structure interaction computations[J]. Journal of Fluids and Structures, 2008, 24(6):819-832.
[42] ALLEN C B. Parallel flow-solver and mesh motion scheme for forward flight rotor simulation:AIAA-2006-3476[R]. Reston:AIAA, 2006.
[43] LIEFVENDAHL M, TRÖENG C. Deformation and regeneration of the computational grid for CFD with moving boundaries:AIAA-2005-1090[R]. Reston:AIAA, 2005.
[44] PERSSON P O, BONET J, PERAIRE J. Discontinuous Galerkin solution of the Navier-Stokes equations on deformable domains[J]. Computer Methods in Applied Mechanics and Engineering, 2009, 198(17):1585-1595.
[45] SHEPARD D. A two-dimensional interpolation function for irregularly-spaced data[C]//Proceedings of the 196823rd ACM National Conference. New York:ACM, 1968:517-524.
[46] WITTEVEEN J A S. Explicit and robust inverse distance weighting mesh deformation for CFD:AIAA-2010-0165[R]. Reston:AIAA, 2010.
[47] WITTEVEEN J A S, BIJL H. Explicit mesh deformation using inverse distance weighting interpolation:AIAA-2009-3936[R]. Reston:AIAA, 2009.
[48] LUKE E, COLLINS E, BLADES E. A fast mesh deformation method using explicit interpolation[J]. Journal of Computational Physics, 2012, 231(2):586-601.
[49] BUHMANN M. Radial basis functions[M]. Cambridge:Cambridge University Press, 2005.
[50] BOER A D, SCHOOT M S V D, BIJL H. Mesh deformation based on radial basis function interpolation[J]. Computers & Structures, 2007, 85(11):784-795.
[51] RENDALL T C S, ALLEN C B. Efficient mesh motion using radial basis functions with data reduction algorithms[J]. Journal of Computational Physics, 2009, 228(17):6231-6249.
[52] RENDALL T C S, ALLEN C B. Reduced surface point selection options for efficient mesh deformation using radial basis functions[J]. Journal of Computational Physics, 2010, 229(8):2810-2820.
[53] SHENG C, ALLEN C B. Efficient mesh deformation using radial basis functions on unstructured meshes[J]. AIAA Journal, 2013, 51(3):707-720.
[54] BOS F M, OUDHEUSDEN B W, BIJL H. Radial basis function based mesh deformation applied to simulation of flow around flapping wings[J]. Computers & Fluids, 2013, 79(6):167-177.
[55] LIU X, QIN N, XIA H. Fast dynamic grid deformation based on Delaunay graph mapping[J]. Journal of Computational Physics, 2006, 211(2):405-423.
[56] 肖天航, 昂海松, 仝超. 大幅运动复杂构型扑翼动态网格生成的一种新方法[J]. 航空学报, 2008, 29(1):41-48. XIAO T H, ANG H S, TONG C. A new dynamic mesh generation method for large movements of flapping-wings with complex geometries[J]. Acta Aeronautica et Astronautica Sinica, 2008, 29(1):41-48(in Chinese).
[57] 周璇, 李水乡, 陈斌. 非结构动网格生成的弹簧-插值联合方法[J]. 航空学报, 2010, 31(7):1389-1395. ZHOU X, LI S X, CHEN B. Spring-interpolation approach for generating unstructured dynamic meshes[J]. Acta Aeronautica et Astronautica Sinica, 2010, 31(7):1389-1395(in Chinese).
[58] LEFRANÇOIS E. A simple mesh defomation technique for fluid-structure interaction based on a submesh approach[J]. International Journal for Numerical Methods in Engineering, 2008, 75(9):1085-1101.
[59] ALAUZET F D R. A changing-topology moving mesh technique for large displacements[J]. Engineering with Computers, 2014, 30(2):175-200.
[60] OLIVIER G, ALAUZET F. A new changing-topology ALE scheme for moving mesh unsteady simulations:AIAA-2011-0474[R]. Reston:AIAA, 2011.
[61] GUARDONE A, ISOLA D, QUARANTA G. Arbitrary lagrangian eulerian formulation for two-dimensional flows using dynamic meshes with edge swapping[J]. Journal of Computational Physics, 2011, 230(20):7706-7722.
[62] ISOLA D. An interpolation-free two-dimensional conservative ALE scheme over adaptive unstructured grids for rotorcraft aerocraft aerodynamics[D]. Milano:Politecnico Di Milano, 2012.
[63] WANG L, PERSSONY P O. A discontinuous galerkin method for the Navier-Stokes equations on deforming domains using unstructured moving space-time meshes:AIAA-2013-2833[R]. Reston:AIAA, 2013.
[64] GOPALAKRISHNAN P, TAFTI D K. A parallel multiblock boundary fitted dynamic mesh solver for simulating flows with complex boundary movement:AIAA-2008-4142[R]. Reston:AIAA, 2008.
[65] ZHOU X, LI S. A new mesh deformation method based on disk relaxation algorithm with pre-displacement and post-smoothing[J]. Journal of Computational Physics, 2013, 235(2):199-215.
[66] LIU Y, GUO Z, LIU J. RBFs-MSA hybrid method for mesh deformation[J]. Chinese Journal of Aeronautics, 2012, 25(4):500-507.
[67] ANDERSON J M, STREITLIEN K, BARRETT D S, et al. Oscillating foils of high propulsive efficiency[J]. Journal of Fluid Mechanics, 1998, 360(1):41-72.
[68] LIAN Y, SHYY W. Aerodynamics of low reynolds number plunging airfoil under gusty environment:AIAA-2007-0071[R]. Reston:AIAA, 2007.
[69] YOUNG J. Numerical simulation of the unsteady aerodynamics of flapping airfoils[D]. Canberra:University of New South Wales, 2005.
[70] THOMAS P D, LOMBARD C K. The geometric conservation law-a link beween finite-difference and finite-volume methods of flow computation on moving grids:AIAA-1978-1208[R]. Reston:AIAA, 1978.
[71] THOMAS P D, LOMBARD C K. Geometric conservation law and its application to flow computations on moving grids[J]. AIAA Journal, 1979, 17(10):1030-1037.
[72] GUILLARD H, FARHAT C. On the significance of the geometric conservation law for flow computations on moving meshes[J]. Computer Methods in Applied Mechanics and Engineering, 2000, 190(11):1467-1482.
[73] GEUZAINE P, GRANDMONT C, FARHAT C. Design and analysis of ALE schemes with provable second-order time-accuracy for inviscid and viscous flow simulations[J]. Journal of Computational Physics, 2003, 191(1):206-227.
[74] FARHAT C, GEUZAINE P. Design and analysis of robust ALE time-integrators for the solution of unsteady flow problems on moving grids[J]. Computer Methods in Applied Mechanics and Engineering, 2004, 193(39):4073-4095.
[75] MAVRIPLIS D J, YANG Z. Construction of the discrete geometric conservation law for high-order time-accurate simulations on dynamic meshes[J]. Journal of Computational Physics, 2006, 213(2):557-573.
[76] HYUNG T A, YANNIS K. Strong coupled flow/structure interaction with a geometrically conservative ALE scheme on general hybrid meshes[J]. Journal of Computational Physics, 2006, 219(2):671-696.
[77] ZHANG L P, WANG Z J. A block LU-SGS implicit dual time-stepping algorithm for hybrid dynamic meshes[J]. Computers & Fluids, 2004, 33(7):891-916.
[78] KAMAKOTI R, SHYY W. Evaluation of geometric conservation law using pressure-based fluid solver and moving grid technique[J]. International Journal of Numerical Methods for Heat & Fluid Flow, 2004, 14(7):851-865.
[79] BOFFI D, GASTALDI L. Stability and geometric conservation laws for ALE formulations[J]. Computer Methods in Applied Mechanics and Engineering, 2004, 193(42):4717-4739.
[80] MASUD A. Effects of mesh motion on the stability and convergence of ALE based formulations for moving boundary flows[J]. Computational Mechanics, 2006, 38(4-5):430-439.
[81] CHANG X H, MA R, ZHANG L P, et al. Further study on the geometric conservation law for finite volume method on dynamic unstructured mesh[J]. Computers & Fluids, 2015, 120:98-110.
[82] 安效民, 徐敏, 陈士橹. 二阶时间精度的CFD/CSD耦合算法研究[J]. 空气动力学学报, 2009, 27(5):547-552. AN X M, XU M, CHEN S L. Analysis for second order time accurate CFD/CSD coupled algorithms[J]. Acta Aerodynamica Sinica, 2009, 27(5):547-552(in Chinese).
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